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Theorem sbcbi2 2887
Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Assertion
Ref Expression
sbcbi2  |-  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) )

Proof of Theorem sbcbi2
StepHypRef Expression
1 abbi 2201 . . 3  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
2 eleq2 2151 . . 3  |-  ( { x  |  ph }  =  { x  |  ps }  ->  ( A  e. 
{ x  |  ph } 
<->  A  e.  { x  |  ps } ) )
31, 2sylbi 119 . 2  |-  ( A. x ( ph  <->  ps )  ->  ( A  e.  {
x  |  ph }  <->  A  e.  { x  |  ps } ) )
4 df-sbc 2839 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
5 df-sbc 2839 . 2  |-  ( [. A  /  x ]. ps  <->  A  e.  { x  |  ps } )
63, 4, 53bitr4g 221 1  |-  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   [.wsbc 2838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2839
This theorem is referenced by: (None)
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